Considering potential attacks from cloud-computing and quantum-computing, it is becoming necessary to provide higher security elliptic curves. The hidden Markov models are introduced for designing the trace-vector computation algorithm to accelerate the search for elliptic curve (EC) base-points. We present a new algorithm for secure Koblitz EC generation based on evolutionary cryptography theory. The algorithm is tested by selecting a secure Koblitz EC over the field F(2 2000 ), with

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... s showing that both the base field and base point of the secure curve generated exceed the parameter range for Koblitz curves recommended by NIST. The base fields generated go beyond 1900 bits, which is higher than the 571 bits recommended by NIST. We also find new secure curves in the range F(2 163 )-F(2 571 ) recommended by NIST. We perform a detailed security analysis of those secure curves, showing that those we propose satisfy the same security criteria as NIST.